Question

Solve by cross multiplication method 6x - 7x +12 =0

Answer 1

Answer:

y substitution, only those equations can be solved, which have unique solution. But the method of cross multiplication discussed below is applicable in all the cases; whether the system has a unique solution, no solution or infinitely many solutions.

Let us solve the following system of equations

a1x + b1y + c1 = 0 ….(1)

a2x + b2y + c2 = 0 ….(2)

Multiplying equation (1) by b2 and equation (2) by b1, we get

a1b2x + b1b2y + b2c1 = 0 ….(3)

a2b1x + b1b2y + b1c2 = 0 ….(4)

Subtracting equation (4) from equation (3), we get

(a1b2 – a2b1) x + (b2c1 – b1c2) = 0

⇒x=b1c2−b2c1a1b2−a2b1

[a1b2−a2b1≠0 and a1a2≠b1b2]

Similarly, y=c1a2−c2a1a1b2−a2b1

These values of x and y can also be written as

xb1c2−b2c1=−ya1c2−a2c1=1a1b2−a2b1

Cross Multiplication Method Examples

Example 1: Solve the following system of equations by cross-multiplication method.

2x + 3y + 8 = 0

4x + 5y + 14 = 0

Sol. The given system of equations is

2x + 3y + 8 = 0

4x + 5y + 14 = 0

By cross-multiplication, we get

Cross Multiplication Method For Solving Equations 1A

⇒x3×14−5×8=x3×14−5×8=12×5−4×3

⇒x42−40=−y28−32=110−12

⇒x2=−y−4=1−2

⇒x2 = −12

⇒ x = – 1

⇒−y−4 = −12

⇒ y = – 2

Hence, the solution is x = – 1, y = – 2

We can verify the solution.